INVARIANT TWINS necessary, although there is a simpler way of reasoning the result out, without actually calculating A, B, C) that J52 - AC = (ps - gr)2 (&" - ac). Now b2 — ac is called the discriminant of the quadratic equation in x\ hence the discriminant of the quadratic in y is B3 — AC> and it has been shown that the discriminant of the transformed equation is equal to the discriminant of the original equation, times (he factor (ps — gr)2 which depends only upon the coefficients p, g, r, s in the transformation y = (px + q)l(rx + s) by means of which x was expressed in terms ofy. Boole was the first (in 1841) to observe something worth looking at in this particular trifle. Every algebraic equation has a discriminant, that is, a certain expression (such as &3 — ac for the quadratic) which is equal to zero if, and only if, two or more roots of the equation are equal. Boole first asked, does the dis- criminant of every equation when its x is replaced by the related y (as was done for the quadratic) come back unchanged except for a factor depending only on the coefficients of the transfor- mation? He found that this was true. Next he asked whether there might not be expressions other 'than discriminants con- structed from the coefficients having this same property of invariance under transformation. He found two such for the general equation of the fourth degree. Then another man, the brilliant young German mathematician, F. M. G. Eisenstein (1823-52) following up a result of Boole's, in 1844, discovered that certain expressions involving both the coefficients and the x of the original equations exhibit the same sort of invariance: the original coefficients and the original x pass into the transformed coefficients and y (as for the quadratic), and the expressions in question constructed from the originals differ from those con- structed from the transforms only by a factor which depends solely on the coefficients of the transformation. Neither Boole nor Eisenstein had any general method for finding such invariant expressions. At this point Cayley entered the field in 1845 with his pathbreaking memoir, On the Theory of Linear Transformations. At the time he was twenty-four. He set himself the problem of firming uniform methods which 429